• Success and Teaching
• Specific Teaching Concerns

Differences between high school and post-secondary courses
High school preparation and transfer of information
Best and worst practices of individual instructors/educational institutions

• Actions to Implement

Reducing impact of differences between high school and post-secondary courses
High school preparation and transfer of information
Best and worst practices of individual instructors/educational institutions

VII. Teaching Recommendations

Best and Worst Practices for High School and Post-Secondary Teaching

Success and Teaching

While the content of the course is the primary focus of the educational process, and the goal of the process is to have students master content, the teacher is the major instrument of the transfer of that knowledge. Although it seems obvious, it is becoming more and more clear that it is necessary to state that teachers of mathematics must be appropriately trained for the level of course they are teaching. Content cannot be taught if it is not mastered by the instructor. Once again, this is a minimal requirement upon the educational community and does not guarantee success; the other pedagogical factors of this section then lead to the successful teaching of mathematics. Each of the aspects of the definition of success – proper background, emphasis on content and learning in the current course, and transfer to other courses and disciplines – impact on the teaching of mathematics.

There are various ways to organize the consideration of the teaching practices and the effects on learning mathematics. This section focuses on separating experiential factors, factors concerning the preparation of students, and factors stemming from the practices of individual instructors/educational institutions. Namely, of concern here are:

• Impediments to learning arising from the differences between the high school and post-secondary environment (experiential factors)
• Importance of a common high school preparation and the transfer of the appropriate content and skills ( preparation factors)
• Best and worst teaching practices of individual instructors/educational institutions

Within this structure, it is important to emphasize the fact that each of the different constituencies in the learning process must be aware of these factors and must address change within each area. It is far too easy for each member of the educational community – students, parents, teachers and administrators – to point fingers at each other rather than recognizing that each member must be fully and appropriately participating in the total process.

Specific Teaching Concerns

Differences between high school and post-secondary courses

Of all disciplines, mathematics has the most structured, continuous flow of concepts and ideas; therefore, it is always amazing that the high school and college classroom experiences can be so different. Primarily, understanding and coping with these differences are up to the student; in most cases, this is appropriate. Due to a variety of reasons, including funding, student maturity, tradition, and market demand, students need to anticipate differences in the college classroom: larger class sizes in introductory/freshman college courses, a pace of presentation that is twice as fast as the high school pace, a higher level of expectations/standards, and a difference in testing patterns. These differences are appropriate to the maturity of the student and the material being presented.

Students are used to having 35 hours a week of structured study and learning time in high school for two semesters; the structure of college, for equivalent work, is one semester of 12 hours a week of structured study and learning time. This means that college faculty, who are preparing material and lectures and grading twice as fast, are expecting students to structure an additional 23 hours per week of studying and learning time on their own. This factor alone is completely missed by students in their planning and is a primary reason for freshman college problems. As a consequence, both inside and outside the classroom, the students are expected to assume significantly more responsibility for their learning. Facts which need to be emphasized repeatedly to students and which deal with these expectations are the following:

1. In post-secondary classes, one can expect larger lecture style classes with a less personal atmosphere.
2. Normally, there are no review days in college classes before or after tests.
3. Comprehensive exams are standard in college and provide a distinctly different challenge from chapter exams. Parents need to understand that comprehensive finals which are fully used in the calculation of grades help prepare the student for college.

High school preparation and transfer of information

Although the high schools deal with a lot of factors over which they have no control in the learning process, they do have some control over the students’ schedules and course content for the time they are in the classroom. Issues which arise in the general education structure and especially impact the learning of mathematics are:

1. Block scheduling: The evidence of the negative impact of block scheduling on the teaching of mathematics is building. In particular,
   
   
• there are fewer minutes devoted to each course (up to 900 fewer minutes),
• the compression of time in which topics are introduced and developed is harmful,
• the amount of daily homework required, and its questionable efficacy, is a heavy volume compared to benefits,
• textbooks are written for 180 days of instruction.
  
  

With less time in class, and that time less effective, teachers report that they are able to teach significantly fewer topics. Algebra I and Advanced Placement calculus are especially disadvantaged by block scheduling, with only 2/3 of the course material being covered in some instances.



2. Timing: Time lapses between mathematics courses are extremely detrimental in high school, in college, and in the transition between the two.
3. Social Passing: Pressure on high school teachers to pass most/all of their students is detrimental in any discipline which requires a continuing progression of ideas.
4. Reading Comprehension: Reading skills inadequate to understand or work with modeling problems hinder the learning of mathematics and the ability to connect mathematics to other disciplines. In particular, it makes "word problems", the analytic and application oriented problems, impossible to handle.
5. Excessive testing: Excessive time spent on testing to address "accountability" or "assessment" issues removes additional time from the learning/teaching process. Most of the testing does not reinforce the current content being studied and therefore does nothing to further the individual student’s learning. This is especially true of tests in a multiple choice format; research shows that multiple choice tests in fact hinder retention and impede affective learning (synthesis and analysis).

Best and worst practices of individual instructors/educational institutions

1. Teacher Preparation: Teachers should be teaching courses which they are trained to teach. Teachers teaching out of their area, even for a short period of time, do irreparable damage to the learning of the students in that class.
2. Teacher Attitude: The teacher must display a positive and enthusiastic attitude towards mathematics and teaching. Additionally, he/she must provide an exceptionally supportive position towards the students.
3. Compressed Courses: Long days of heavily compressed material, such as regularly occurs in Summer School, can impede retention and comprehension of mathematics. Mathematics requires a deliberate pace, with sufficient time to establish a base of knowledge before building upon it.
4. Class size: Large classes, with impersonal environments and an over-dependence on multiple choice testing, hinder learning.
5. Teaching Practices: Wherever you stand on the "traditional-reform" teaching spectrum, there are certain "Worst Practices" which occur all too often due to circumstances or institutional policy. Some of these negative practices can be characterized:
   
   
• Wasting valuable class time (for example, long undirected activity sessions, and spending inordinate amounts of time reviewing previously covered material.)
• Insufficient analysis of hands-on activities to determine student learning;
• Exclusive use of technology to visualize the most basic processes and functions; i.e., some things ought to be remembered;
• Inappropriate use of partial credit: too little discourages students while too much gives students a false sense of the requirements and of their level of knowledge;
• Presentation and explanation of material from a single perspective.

Actions to Implement

Reducing impact of differences between high school and post-secondary courses

1. Comprehensive Exams: Comprehensive exams must be given in high school. This improves high school as well as college performance in mathematics. The committee recommends that a comprehensive exam, with a weight of 25% of the course grade, be given at the end of every high school mathematics course. The following benefits arise from this practice:
   
   
• Enables students to connect and integrate the topics of the course.
• Holds students accountable for all of the material of the course.
• Prepares students for the types of tests they will encounter both in college and in their career.
• Encourages basic study skills: note-taking, good organization, and test-taking skills.
• Encourages an in-depth review of the course – a valuable "second look" in the context of the course as a whole.
• Allows student to see the "big picture" – major concepts and unifying themes.
  
  
2. Office Hours: Students in college need to be encouraged to take advantage of office hours and other resources provided by the post-secondary institution.
3. Study groups: Students need to set up their own study groups and meet with them once a week outside class hours.
4. Independent Review: Students need to design their own review strategies, with the help of the instructor during office hours if necessary, to prepare for tests. It is also the student’s responsibility to learn from graded work individually or in discussion with the instructor outside of class.

High school preparation and transfer of information

1. Block scheduling: School districts must consider creative scheduling solutions that will address these concerns; in particular, AP calculus should be scheduled for a full year. Additionally, appropriate textbooks, when they become available, must be adopted to accommodate the longer sessions with less total time available.
2. Timing of Courses: We recommend that mathematics be taken all four years of high school and that college students take mathematics in their first semester to ensure continuity of learning and the maximum retention of high school material.
3. Course Scheduling: We recommend that the four-year sequence be Algebra I, Algebra II, Geometry and Pre-Calculus, in that order. Furthermore a five-year sequence, ending with AP Calculus, is a tremendous advantage toward post-secondary success with mathematics.
4. False learning: There are standard mistakes made throughout algebra and calculus which are not being corrected. The classic problems arise in the following areas:
   
   
• Lack of estimation skills and a sense of whether an answer is correct or not
• No competence with fractions; can only do arithmetic with decimals on a calculator
• Excessive written calculations which could and should be done mentally
• Insufficient explanation of the process used when writing down a solution
• Canceling terms instead of using factorization
• 
• Solving inequalities without proper attention to negatives, positives, and absolute value.

Best and worst practices of individual instructors/educational institutions

1. Teacher Preparation: High school mathematics teachers should have the equivalent course work for a major in mathematics at the Bachelor of Science or Bachelor of Arts level. In post-secondary settings, no teacher should be teaching who does not have the above credentials plus hours towards a masters in mathematics.
2. Teacher Assignments: Unprepared teachers arise from courses being assigned at the last minute. This practice should be kept at a minimum if it cannot be eliminated completely.
3. Teacher Attitude: The teacher must have working conditions and preparation time which allows a positive attitude towards mathematics and towards the students. When the proper support is provided by the institution, then the teacher can begin to affect the student attitudes.
4. Summer School: Students should be strongly discouraged from taking remedial or repeat coursework in the summer in high school and in the first two years of college.
5. Class size: A high school class size of less than 25 is appropriate; larger classes hinder learning. Although there is no universal standard applied in college courses regarding class size, the class size needs to allow for effective instruction at the particular institution.
6. Teaching Practices: Negative-teaching practices can be overcome through appropriate preparation and training. Teachers who follow the best practices often exhibit the following attributes:
   
   
• Clear teacher knowledge and competence
• Visible teacher enthusiasm and excitement
• Appropriate notification of teaching assignments
• Teaching assignments made which are appropriate to teacher’s educational training
• Technology used to enhance concept development
• Homework and teaching emphasize the numeric, graphic, and analytic characteristics of the material
• Good pacing of material – correct balance to between "covering" the necessary material and responding to student needs; avoiding a rush to catch up towards the end
• Good rapport and interactions with students – enough flexibility;
• Use of manipulatives to enhance concepts rather than as time-consuming, distracting gimmicks
• Develop high but realistic expectations of the students
• Assist students in setting realistic goals for themselves
• Committed to continuing professional development, especially in content
• Design teacher-directed lessons with active involvement by both student and teacher for the full class period.
  
  
7. Middle School: There should be a statewide implementation of a middle school certification in mathematics with identification of an appropriate body of knowledge.

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Last updated: May 11, 1999
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